MATRIX ELEMENTS LATTICE 2001

Theoretical and Numerical

Results after Lattice 2000

(only light quarks)

Guido Martinelli

(Special thanks to D. Becirevic, M. GoltermanR. Gupta, D. Lin, R. Mawhinney,J. Noaki and M. Papinutto, S. Sharpe)

‹ N' |Q(0)|N› Q= q q , q 5 q , q q, q DDD …q

‹ K0 | Q S=2 | K0 › & ‹ π | Q i | K ›

‹ K0 | Q S=2 | K0 ›, ‹ π | Q i | K › &

‹ π π | Q i | K › +chiral expansion

‹ 0 | Q (0)| π › Q= q 5 q , q 5 DDD …q

Perturbative vs Non-perturbative vs Ward Identities, Scaling etc.

Non-improved, Improved, Twisted,Anisotropic Lattices etc.

In a fixed gauge, gauge invariant etc.

Total = 18 + 20 presentations + 13 posters ! one talk

see the accurate and complete reviews by

S2 = S. Sint and L2 = L. Lellouch

at Lattice 2000

A Selection has been unavoidable

Apologizes to the authors of submitted contributions and the speakers of talks given in the

parallel sessions which I had to omit in my review

Many thanks to all my collegues who have kindly sent material and information for the preparation of

this talk

Plan of the talk

• A few physics issues (not on the lattice);• The UV problem: results on non-perturbative renormalization (heavy-light not covered see talk by S. Ryan); perturbative; here and there• The IR problem: non-leptonic weak decays and related items;• Physics issues for the lattice (see also talk by M. Beneke); here and there• Conclusions and outlook.

Exp Th

2.271 0.017 10-3 (1-) BK

' / 17.2 1.8 10-4 -7 ÷ 30 10-4

( RBC AND CP-PACS to be discussed in the next slide )

Ms / Md >30 (95%cl) [(1- )2 + 2]-1 see talk by S. Ryan

BR(B Xs ) 3.11 0.39 10-4 3.50 0.50 10-4 BR(K+ + ) 1.5 +3.4-1.2 10-10 0.8 0.3 10-10

no lattice QCD needed figures

Observed Genuine FCNC

Physics Results from RBC and CP-PACSno lattice details here, they will be discussed below. See talks by

Mawhinney,Calin,Blum and Soni (RBC) and Noaki (CP-PACS)

Re(A0) Re(A2) Re(A0)/Re(A2)

′/

RBC 29÷3110 -8

1.1 ÷1.210 -8

24÷27 -4 ÷ -810 -4

CPPACS

16÷2110 -8

1.3÷1.510 -8

9÷12 -2 ÷ -710 -4

EXP 33.310 -8

1.5 10 -8 22.2 17.2 ±1.810 -4

Total Disagrement with experiments ! (and other th. determinations)

Opposite sign !

New Physics?

0.5

1.0

1.5

2.0

2.5

3.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

B6

B8

'/ ~ 13 (QCD/340 MeV)Im t (110 MeV/ms )[B6 (1-IB ) -0.4 B8 ]

'/ =0

DonoghueDe Rafael

Artistic representation of present situation

Chromomagnetic operators vs '/ and

Og = g (sL

ta dR Ga sR

ta dL Ga )

16 2

H g = C+g O+

g + C-g O-

g

• It contributes also in the Standard Model (but it is chirally supressed mK4)

• Beyond the SM can give important contributions to ' (Masiero and Murayama)• It is potentially dangerous for (Murayama et. al., D’Ambrosio, Isidori and G.M.)• It enhances CP violation in K decays (D’Ambrosio, Isidori and G.M.)

• Its cousin O gives important effects in KL 0 e + e-

( ‹ π0 | Q + | K0 › computed by D. Becirevic et al. , The SPQcdR Collaboration,

Phys.Lett. B501 (2001) 98)

LCP = LF=0 + L F=1 + L F=2

F=0 de < 1.5 10-27 e cm dN < 6.3 10-26 e cm

F=1 ' /

F=2 and B J/ Ks

After the first attempts at the end of the 80s (Aoki, Manohar, Sharpe, Gocksch)the calculation of the matrix element of the neutron electric dipole moment has been abandoned. Renormalization of this operator and calculation of disconnecteddiagrams with stocastic sources is now a common practice

Important for :• Strong CP problem u 5 u + d 5 d or Ga Ga

• SUSY extention of the Standard Model

di di di di di di

(C+j)C (C+

j)C

U+k

D-k D-

k

N0j

D-k D-

k

ga

LF=0 = -i/2 Ce 5 F

-i/2 CC 5 ta Ga

-1/6 Cg fabc Ga G b

Gc

Ce,C,g can be computed perturbatively

LSMF=2 = ij=d,s,b (Vtdi

V*tdj

)2 C [di (1- 5 ) dj] 2

LF=2general

= ij=d,s,b (Vtdi V*

tdj)2 C ij Q ij

= different Lorentz structures LL, L R etc.C ij =complex coefficients from perturbation theory

K I Q ij I K from lattice QCD (Donini et al.

Phys. Lett. B470 (1999) 233; phenomenological analyses Ciuchini et al.; Ali and London; Ali and Lunghi; Buras et al.; Bartl et al.)

With/Without subtractions presented at this Conference by Becirevic,SPQcdR Collaboration (also B I Q ij I B )

NEW RESULTS FOR BK slide I

World Average by L.Lellouchat Lattice 2000 0.63 0.04 0.10 0.86 0.06 0.14

CP-PACS perturbative renorm. 0.575 0.006 0.787 0.008(quenched) DWF 0.5746(61)(191)

RBC non-perturbative renorm. 0.538 0.008 0.737 0.011(quenched) DWF

SPQcdR (preliminary) 0.71 0.13 0.97 0.14(quenched) Improvedwith subtractions

without subtractions 0.70 0.10 0.96 0.14

BNDRK(2 GeV) BK

=6.2 non-perturbatively improved actiojn

Some questions on BK

BK computed by De Rafael and Peris in the chirallimit is very small BK = 0.38 0.11

Is this due to chiral corrections ?BK /BK

chiral = 1.10(8) (with subtractions)

1.11(10) (without subtractions)Becirevic quenched

A large value of BK ~ 0.85 corresponds to a toolarge value of K+ π+ π0 if SU(3) symmetryand soft pion theorems at lowest order are used (J. Donoghue 82) . Even 0.75 is too large.RBC and CP-PACS seems to find instead the physicalK+ π+ π0 amplitude. What is their value for BK /Bkchiral ?

figure

NEW RESULTS FOR BK slide II

K0 | Q S=2 | K0 with Wilson-Like Fermions without subtractions

of wrong chirality operators. Two proposals with the same physical idea Q VV+AA QVA which cannot mix because of CPS

Talk by D. Becirevic at this Conference (D. Becirevic et al. SPQcdR )Use CPS and Ward ids, only exploratory results at Latt2000 New numerical results (with NPR on quark states taking into account theGoldstone Boson Pole, see the talk by C. Dawson at Latt2000, see also Pittori and Le Yaouanc)

Talk by C. Pena at this Conference (Guagnelli et al., ) Use tmQCD and Schrödinger Functional Renormalization, only a proposal at Latt2000 numerical results for the bare operator BK(a)= 0.94(2) on V=16332 and 0.96(2) (at a given value of thequark masses) on V=16348 The SF renormalization is underway

With subtractionsRI-MOM

Without subtractions

Chiral behaviour of K0 | Q S=2 | K0 talk by D. Becirevic SPQcdR= 6.2 V=24364 200 configurations

Chiral behaviour of K0 | Q S=2 | K0 talk by Pena

Tm FermionsChiral Behaviourneeds further study

NEW RESULTS FOR BK slide III

Sinya Aoki at this Conference presented preliminary results on therenormalization of bilinear quark operators with the SF methodand Domain Wall Fermions. The plan is to extend these calculationsto S=2 operators

A forgotten method: Rossi, Sachrajda, Sharpe, Talevi, Testa and G.M.,Phys. Lett. B411 (1997) 41 (easy to implement, gauge invariant, no contact termssee also G.M. NPB(PS) 73 (1999)58. )

Z2J T[J(x)J† (0)] | |x|=1/ << 1/QCD

= T[J(x)J† (0)] tree

1/a << 1/|x| ˜ << 1/QCD window avoided by iterative

matching of the renormalization scale at different values of

T[J(x) J† (0)] | |x| << 1/QCD, perturbative=

The necessary two loop calculations for bilinear operators have been completed (Del Bello and G.M.).

J J

S

Q

The extension two four Fermion operators is straightforward

J J

J J

Perturbative calculations with overlap fermions by S. Capitani and L. Giustihep-lat/0011070 v2 see also S. Capitani and L. Giusti Phys. Rev. D62 (2000) 114506See talk by Capitani in the parallel session

Perturbative calculations for DWF by S. Aoki et al. , Phys. Rev. D59 (1999) 094505,Phys. Rev. D60 (1999) 114504, Phys. Rev. D63 (2001) 054504 and in preparation

Renormalization of Four Fermion Operators

Renormalization & Improvement

Very nice and complete analysis of improvement constants by Bhattacharya

Gupta Lee and Sharpe cA, bA - bV, cT etc. at =6.0,6.2 and 6.4

IMPROVEMENT OF FOUR FERMION OPERATORS ?

figure

I=1/2 and ′/

• I=1/2 decays (Q1 and Q2)

• ′/ electropenguins (Q7 and Q8)• ′/ strong penguins (Q6)

• K π π from K π and K 0

• Direct K π π calculation

Theoretical Novelties

• Chiral Perturbation Theory for ‹ Q +,1,2,7,8› V. Cirigliano

and E. Golowich Phys. Lett. B475 (2000) 351; M. Golterman and E. Pallante JHEP 0008 (2000) 023; talks by D. Lin and E. Pallante at the parallel session.

• FSI and extrapolation to the physical point Truong, E. Pallante and A. Pich (PP) Phys. Rev. Lett. 84 (2000) 2568; see also A. Buras at al. Phys. Lett. B480 (2000) 80 talk by G. Colangelo

• ‹ π π IQ i I K › on finite volumes L. Lellouch

& M. Luscher Commun. Math. Phys. 219 (2001) 31 (LL) and D.Lin,

G.M., C. Sachrajda and M. Testa hep-lat/0104006 (LMST)

Only the subjects with a will be discussed

New Numerical Results

Chiral behaviour of ‹ π π | Q 4 | K ›;

First determination of ‹ π π | Q 7,8 | K › and of their chiral behaviour;

First signal for ‹ π π | Q 1,2 | K › and ‹ π π | Q 6 | K ›;Gladiator The SPQcdR Collaboration

(Southapmton, Paris, Rome,Valencia)

results presented by D. Lin and M. Papinutto

‹ π | Q i | K › for I=1/2 and ′/ with domain wall fermions

CP-PACS talk by Noaki

RBC talks by Mawhinney,Calin,Blum and poster by Soni

VicenteGimenez

GENERAL FRAMEWORK

HS=1 = GF/√2 Vud Vus*[ (1-) i=1,2 zi (Qi -Qc

i) +

i=1,10 ( zi + yi ) Qi ] Where yi and zi are short distance coefficients, which are knownIn perturbation theory at the NLO (Buras et al. + Ciuchini et al.)

= -Vts*Vtd/Vus

*Vud

We have to compute AI=0,2i= ‹ (π π)I=0,2 IQ i I K ›

with a non perturbative technique (lattice, QCD sum rules, 1/N expansion etc.)

AI=0,2i () =‹ (π π)I=0,2 IQ i ()I K ›

= Zik( a) ‹ (π π)I=0,2 IQ k (a)I K ›Where Q i (a) is the bare lattice operatorAnd a the lattice spacing. Two main roads to the calculation:

• K π π from K π and K 0• Direct K π π calculation

So far only qualitative (semi-quantitative) results for ‹ (π π)I=0 | Q 1,2,6 | K ›

from Lattice QCD

Main sources of systematic errors from the UV and IR behaviour of the theory

UV: In order to obtain the physical amplitude we need Zik( a).

The construction of finite matrix elements of renormalized operators from the bare lattice ones is in principle fully solved

C. Bernard et al. Phys. Rev. D32 (1985) 2343.M. Bochicchio et al. Nucl. Phys. B262 (1985) 331;L. Maiani et al. Nucl. Phys. B289 (1987) 505; C. Bernard et al. Nucl. Phys. B (Proc. Suppl.) 4 (1988) 483;C. Dawson et al. Nucl. Phys. B514 (1998) 313.S. Capitani and L. Giusti hep-lat/0011070.

Several non-perturbative techniques have been developed in order to determine Zik( a). The systematic errors can be as small as 1% for quark bilinears and typically (so far) 10% for four fermion operators. For Q1,2,6 only perturbative calculations (error 20-25%) so far (but see RBC Collaboration, C. Dawson et al.

Nucl.Phys.Proc.Suppl.94:613-616,2001) . More work is needed !!

Discretization errors are usually of O(a), O(mqa) or O(|p|a), but can become of O(a2) with Domain Wall Fermions or Non-perturbatively improved actions and operators. Similar problems encountered in effective theories with a cutoff (see V. Cirigliano, J. Donoghue, E. Golowich JHEP 0010:048,2000 hep-ph/0007196 )

The IR problem arises from two sources:• The (unavoidable) continuation of the theory to Euclidean space-time (Maiani-Testa theorem) • The use of a finite volume in numerical simulations

An important step towards the solution of the IR problem has been achieved by L. Lellouch andM. Lüscher (LL), who derived a relation between

the K π π matrix elements in a finitevolume and the physical amplitudes

Commun.Math.Phys.219:31-44,2001 e-Print Archive: hep-lat/0003023

Here I discuss an alternative derivation based on the behaviour of correlators of local operator when V D. Lin, G.M., C. Sachrajda and M. Testa hep-lat/0104006 (LMST)

presented by L. Lellouch at Latt2000

Consider the following Euclidean T-products (correlation functions) G(t,tK)= ‹ 0 | T [J(t) Q(0) K+ (tK)] |0 ›, G(t) = ‹ 0 | T [J(t) J(0) ] |0 ›, GK (t) = ‹ 0 | T [K(t) K+(0) ] |0 ›,

where J is a scalar operator which excites (annhilates) zero angular momentum () states from (to) the vacuum and K is a pseudoscalar source which excites a Kaon from the vacuum (t > 0 ; tK < 0)

G(t,tK) G(t) GK (t)

J(t)

Q

K+(tK) J+(0)

J(t) K+(0) K(tK)

K

At large time distances:G(t,tK) V n ‹ 0 |J | n›V ‹ n |Q(0) |K ›V ‹ K |K+ |0 ›V

exp[ -(Wn t +mK| tK |) ] G(t) = V n ‹ 0 |J | n›V ‹ n|J | 0›V exp[ -Wn t ]

From the study of the time dependence of G(t,tK), G(t) and GK (t) we extract • the mass of the Kaon mK • the two- energies Wn

• the relevant matrix elements in the finite volume‹ K |K+ |0 ›V , ‹ 0 |J | n›V , and ‹ n |Q(0) |K ›V

We may also match the kaon mass and the two pion energy, namely to work with mK = Wn* Necessary to obtain a finite I=1/2 matrix element

The fundamental point is that it is possible to relate the finite-volume Euclidean matrix element with the absolute value of

the Physical Amplitude |‹ E |Q(0) |K ›| by comparing, at large values of V, the finite volume correlators to the infinite volume ones

|‹ E |Q(0) |K ›| = √F ‹ n |Q(0) |K ›V

F = 32 2 V2 V(E) E mK/k(E) where k(E) = √ E2/4- m2

and

V(E) = (q ’(q) + k ’(k))/4 k 2 is the expression which one would

heuristically derive by interpreting V(E) as the density of states in a finite volume (D. Lin, G.M., C. Sachrajda and M. Testa)

On the other hand the phase shift can be extractedfrom the two-pion energy according to (Lüscher):

Wn = 2 √ m2 + k2 n - (k) = (q)

• The LL formula is derived at fixed finite volume (n < 8) whereas the LMST derivation holds for V at fixed energy E;• It is possible to extract the matrix elements even when mK Wn* this is very useful to study the chiral behaviour of ‹ |Q(0) |K ›

• In the near future, in practice, it will only be possible to work with a few states below the inelastic threshold

G(t,tK) = V n ‹ 0 |J | n›V ‹ n |Q(0) |K ›V ‹ K |K+ |0 ›V exp[ -(Wn t + mK| tK |) ]

For the validity of the derivation, inelasticity at Wn* must be small (which is realized for

states with Wn* = mK);

• If one uses G(t1, t2,tK) = ‹ 0 | T [ (t1) (t2) Q(0) K+ (tK)] |0 ›, no correcting factor is necessary; in this case we get the real part of the amplitude R= |‹ E |Q(0) |K ›| cos (E) + O(1/L)

Main differences between LL and LMST:

Wn is determined from the time dependence of the correlation functions

G(t,tK) = V ‹ K |K+ |0 ›V exp( -mK| tK | ) n ‹ 0 |J | n›V ‹ n |Q(0) |K ›V exp (- Wn t ) = n A n exp (- Wn t )

From Wn it is possible to extract the FSI phase (for a different method to obtain (E) =(k) see LMST)

Wn = 2 √ m2 + k2 n - (k) = (q)

1) IT IS VERY DIFFICULT TO ISOLATE Wn WHEN n IS LARGE !

2) THIS METHOD HAS BEEN USED FOR THE I=3/2 AND 1/2 TRANSITIONS DISCUSSED IN THE FOLLOWING and talks by Lin and Papinutto

We can work with Wn* mK at several values of the pion masses and momenta (and at different kaon masses) and extrapolate to the physical point by fitting the amplitude to its chiral expansion, including the chiral logarithms. Two extra operators needed with respect to Pallante and Golterman. This is underway for Q4 and the electropenguins.

Example: -i M4 = [mK(m +E)/2+ m E ] + 4 2 m

2 (m2 - mK

2 ) + 4mKm (m2 +mK

2 )+ …. (E =√ m

2 + p2 )

In general M4 =M4 (mK , m ,E )

Summary of the main steps

1) Extraction of the signal yes yes (NEW !!)

2) Renormalization non pert

3) Chiral extrapolation to the yes not yet physical point (possible with more statistics)

4) Discretization errors not yet not yet (possible in (possible in the near future) the near future)

5) Quenching possible in possible near future

K π π I=3/2 I=1/2for ‹ K0 | Q S=2

| K0 › 1)-5)

pert !!

time

time

‹π π IQ 4 I K ›

‹π π IQ 8 I K ›

Present statistical error of O(10%)

Future statistical error < 3%

I = 3/2THE SIGNAL: Improved action350 Configurations = 6.0 ( a-1=2 GeV)

THE CHIRAL BEHAVIOUR FOR ‹π π IQ 4 I K ›

for the chiral behaviour of ‹Q 4› see for example Pallante and Golterman and Lin;chiral logs and extra operators not yet included; cos (E) ≈ 1

NEW !! THE CHIRAL BEHAVIOUR FOR ‹π π IHW I K ›I=2 and a comparison with JLQCD Phys. Rev. D58 (1998) 054503 (non improved perturbative renormalization) & experiments

Aexp= 0.0104098 GeV3

This work 0.0097(10) GeV3

JLQCD 0.009(2)÷0.011(2) GeV3

Lattice QCD finds BK = 0.86 and a value of ‹π π IHW I K ›I=2 compatible with exps

THE CHIRAL BEHAVIOUR FOR ‹π π IQ 8I K ›I=2

for ‹Q 7,8› formulae by V. Cirigliano and E. Golowich

RI-MOM renormalization scheme

Results for Q 7,8 and comparison with other determinations (MS)

K ’ π π(SPQcdR) NEW!!

0.53 ± 0.06 0.02 ± 0.01

J . Donoghu e and. E Golowich

1.3 ± 0.3 0.22 ± 0.05

M. Knecht, .SPeri s a . nd EDeRafael

3.5 ± 1.1 0.11 ± 0.03

Donini e t a.l(Rome). D Becirevi c e tal.

(SPQcdR) NEW!!

0.5 ± 0.10.49 ± 0.06

0.11 ± 0.040.10(2)(1)

<Q 8> <Q 7>

from K π

GeV3

RBC and CPPACS for comparison

~ 2.2

I = 1/2 and '/ The subtractions of the power divergencies, necessary to obtain finite matrix elements, are the major obstacle in lattice calculations

1) these subtractions are present for both the methods which have been proposed K π π from K π and K 0 Direct K π π calculation

2) the subtractions are not needed for ‹π π IQ 4,7,8 I K › I=2

Example I = 1/2 from K π π

Q = [ s (1-5) d u (1-5) u s (1-5) u u (1-5)d ] - [ c u ] this is the subtraction !!

I = 1/2 and '/ The most important contributions are

expected to come from penguin diagrams

Lattice calculations have shown that it is non possible to explain the octetenhancement with emission diagrams only; penguins are at the origin of thePower divergences. They are absent in I=3/2 amplitudes.

Disconnected emission Disconnected Penguin

B1,2 ≈ 4,5

B6 ≈ 3

Matrix element of ‹π π IQ-I K ›I=0 without Z(a) only penguin contractions with GIM subtractions

1) Data with 340 confs2) Statistical error 50÷70% 3) Needed about 5000 confs for an error of 20 % (quenched)4) Actually about 20 confs/day ( 9 months)6) With further improment of the programmes and the 3rd machine45/day (4 months)

Signal

For bare ops ‹π π IQ-I K ›I=0/ ‹π π IQ+I K ›I=2 ≈ 9 !!

mK = 2 m

Matrix element of ‹π π IQ6I K ›I=0 without Z(a)

only penguin contractions

Signal

mK = 2 m

No subtraction needed ‹π π Is 5 dI K › = ‹π π I s 5 dI K ›/(ms+md)= 0

‹π π Is 5 dI K ›

‹π π IQ6I K ›I=0

See C. Dawson et al. Nucl. Phys. B514 (1998) 313

K π π from K π and K 0and Domain Wall Fermions

• Impossible (in practice) with Wilson (Improved Fermions) because of (power) subtractions;

• With Domain Wall Fermions (DWF) only one subtraction is required (see below); Also true with Overlap Fermions

• Computer much more demanding than Wilson or Improved Fermions thus only K π so far;

• A very good control of residual chiral symmetry breaking is

required. The error decreases as exp(- const. L5 ) but remember that we have power divergences;

• Problems with the extrapolation to the physical point (Pich Pallante see also talk by Colangelo) ??

A subtraction is needed: ‹π I Qi

sub I K › = ‹π I Qi

sub I K ›-ci (ms + md) ‹π I s d I K ›;

ci obtained from the condition

‹0I Qi I K › - ci (ms - md) ‹0I s 5 d I K › = 0;

ci is obtained either using non degenerate quarks (RBC) or from the derivative of the 2-point correlation function (CP-PACS)

C. Bernard et al. Phys. Rev. D32 (1985) 2343.

Preliminary numerical results for all the operators were presented by CP-PACS and RBC) at Lattice 2000; Physics Results at Lattice 2001.

RBC Standard GaugeField Action =6.0 DWF

RBC Standard GaugeField Action =6.0 DWF

Linear fit ??

O6

O6

CP-PACS RG-improved GaugeField Action =2.6 DWF

RBC Standard GaugeField Action =6.0 DWF

O2

O2

CP-PACS RG-improved GaugeField Action =2.6 DWF

Physics Results from RBC and CP-PACStalks by Mawhinney,Calin,Blum and Soni (RBC)

Noaki (CP-PACS) Re(A0) Re(A2) Re(A0)/

Re(A2)′/

RBC 29÷3110 -8

1.1 ÷1.210 -8

24÷27 -4 ÷ -810 -4

CPPACS

16÷2110 -8

1.3÷1.510 -8

9÷12 -2 ÷ -710 -4

EXP 33.310 -8

1.5 10 -8 22.2 17.2 ±1.810 -4

• Chirality• Subtraction• Low Ren.Scale• Quenching • FSI• New Physics• A combination ?

Even by doubling O6 one cannot agree with the dataK π π and Staggered Fermions (Poster by W.Lee) will certainly help to clarify the situation I am not allowed to quote any number

Unphysical quenched contributions I = 1/2 and '/ Golterman and Pallante

presented by Pallante Q6 is an (8,1) operator. In the quenched case it may mix with an (8,8) operator , as it can be explicit checked in one loopquenched chiral perturbation theory

This correction is potentially more importantfor Q6 since

GP suggested to remove q q contractions inEye/Ann Diagrams to get rid of the unphysical(8,8) contributionsTHE COMPARISON GIVES US AN INDICATION OF THE SYST. ERROR

‹π I Q 6 I K › ~ m2

‹π I (8,8) I K › ~ 1 thus, the one

loop correction ‹π I Q (8,8) 6 I K › ~ m2

Figure

A TESTING GROUND FOR K π π CALCULATIONS

J. DONOGHUE AT KAON 2001

study K π π l l namely

‹ S=0, I=0 | u (1- 5 )s |K ›

Simple case for Maiani-Testa theoremRenormalization trivial (no mixing no power div.)Chiral expansion known at 2 loops

Conclusions and OutlookMANY PROGRESSES

1) The possibility of computing the physical K π π amplitude has been demonstrated by LL (see also LMST);

2) For the first time there is a signal for K π π penguin-like contractions of Q1,2,6 . More work is needed to reduce the uncertainties (statistical and systematic);

3) The new results with Domain Wall Fermions for K π amplitudes are really puzzling;

4) The chiral extrapolation to the physical point (quenched, unquenched, infinite and finite volumes) is critical;

4) The extension of LL/LMST to non-leptonic B-decays (e.g. B K π), for which the two light mesons are above the inelastic threshold, remains an open problem worth being investigated.

… follow Martin Lüscher suggestion, small and smart is often better than big and ….

David and Golia by Caravaggio